You should not have to pick up your luggage, as your stopover is less than 24 hours.

If you elect to pick up your luggage, and go from T2 to T3, you will have to go through customs, etc, which takes much more time than a direct transit(never enter UK). At least an hour, probably less than 2.

Heathrow really shuts down at night. If you are planning to move luggage from LH to SQ(which I don't think you have to do), make certain the SQ booth is attended at that hour.